Aleksey D. Drozdov

Model for Anomalous Moisture Diffusion through a Polymer-Clay Nanocomposite

Experimental data are reported on moisture diffusion and the elastoplastic response of an intercalated nanocomposite with vinyl ester resin matrix and montmorillonite clay filler at room temperature. Observations in diffusion tests showed that water transport in the neat resin is Fickian, whereas it becomes anomalous (non-Fickian) with the growth of the clay content. This transition is attributed to immobilization of penetrant molecules on the surfaces of hydrophilic clay layers. Observations in uniaxial tensile tests demonstrate that the response of vinyl ester resin is strongly elastoplastic, whereas an increase in the clay content results in a severe decrease of plastic strains observed as a noticeable reduction in the curvatures of the stress-strain diagrams. This is explained by slowing down the molecular mobility in the host matrix driven by confinement of chains in galleries between platelets. Constitutive equations are developed for moisture diffusion through and the elastoplastic behavior of a nanocomposite. Adjustable parameters in these relations are found by fitting the experimental data. Fair agreement is demonstrated between the observations and the results of numerical simulation. A striking similarity is revealed among changes in diffusivity, ultimate water uptake, and the rate of plastic flow with an increased clay content.

A micro-mechanical model for the response of filled elastomers at finite strains

Abstract Constitutive equations are developed for the isothermal response of particle-reinforced elastomers at finite strains. A rubbery polymer is treated as a network of chains bridged by junctions. A strand between two junctions is thought of as a series of inextensible segments linked by bonds. Two stable conformations are ascribed to a bond: flexed and extended. Deformation of a specimen induces transition of bonds from their flexed conformation to the extended conformation. A concept of trapped entanglements is adopted, according to which not all junctions are active in the stress-free state. Under straining, some entanglements are transformed from their passive (dangling) state to the active state, which results in a decrease in the average length of a strand. Stress–strain relations for an elastomer and kinetic equations for the rate of transition of bonds from their flexed conformation to the extended conformation are derived by using the laws of thermodynamics. Simple phenomenological equations are suggested for the evolution of the number of active entanglements. The model is determined by five adjustable parameters which are found by fitting experimental data in uniaxial tensile tests. Fair agreement is demonstrated between the results of numerical simulation and observations for a polysulfide elastomer reinforced with polystyrene particles and two natural rubber vulcanizates with different cross-linkers.

A constitutive model for nonlinear viscoelastic media

Abstract A new constitute model is derived for the nonlinear viscoelastic behavior of polymers under isothermal loading. The model extends the concept of adaptive links (entanglements) between polymeric molecules to aging viscoelastic solids. By using data for polypropylene and polyurethane in relaxation tests, material parameters are found. To verify the model, experimental data for loading/unloading with constant rates of strains are employed, which demonstrate fair agreement between measurements and their prediction. The model is utilized to calculate stresses and displacements built-up in a conic pipe under the action of torques applied to its edges. An explicit solution to this problem is derived, where a time-dependent coefficient satisfies a nonlinear integral equation. The effect of geometrical parameters of the pipe and the loading history on stresses and displacements is studied numerically. Two correspondence principles are derived which permit a solution to a viscoelastic problem to be found explicitly in terms of solutions to appropriate elastic problems. Some restrictions on the model nonlinearity are determined which imply these principles.

The effect of annealing on the elastoplastic response of isotactic polypropylene

Abstract Four series of tensile loading–unloading tests are performed on isotactic polypropylene in the sub-yield domain of deformations at room temperature. In the first series, injection-molded specimens are used as produced, whereas in the other series the samples are annealed for 24 h at 120, 140 and 160 °C, which covers the low-temperature region and an initial part of the high-temperature region of annealing temperatures. A constitutive model is developed for the elastoplastic behavior of a semicrystalline polymer. The stress–strain relations are determined by five adjustable parameters that are found by fitting the experimental data. The effect of annealing is analyzed on the material constants.

Non-linear viscoelasticity and viscoplasticity of isotactic polypropylene

Abstract Observations are reported in tensile tests with constant cross-head speeds (ranging from 5 to 200 mm/min), relaxation tests (at strains from 0.02 to 0.08), creep tests (at stresses from 15.0 to 25.0 MPa) and recovery tests (after straining up to the maximal strains ranging from 0.04 to 0.12 and subsequent retraction) on isotactic polypropylene at room temperature. A constitutive model is derived for the time- and rate-dependent responses of a semicrystalline polymer at isothermal deformation with small strains. A polymer is treated as an equivalent heterogeneous network of chains bridged by temporary junctions (entanglements, physical cross-links and lamellar blocks). The network is thought of as an ensemble of meso-regions linked with each other. The viscoelastic behavior of the ensemble reflects thermally-induced rearrangement of strands (separation of active strands from temporary junctions and merging of dangling strands with the network). To describe the viscoplastic response, the entire plastic deformation is split into the sum of two components: one of them is associated with sliding of junctions in the non-affine network of chains, while the other accounts for coarse slip and fragmentation of lamellar blocks. Stress–strain relations and kinetic equations for the plastic strains are developed by using the laws of thermodynamics. The constitutive equations involve five material constants that are found by fitting the observations. Fair agreement is demonstrated between the experimental data and the results of numerical simulation.

Fractional differential models in finite viscoelasticity

SummaryA new class of constitutive models is derived for viscoelastic media with finite strains. The models employ the so-called fractional derivatives of tensor functions.We introduce fractional derivatives for an objective tensor which satisfies some natural assumptions. Afterwards, we construct fractional differential analogs of the Kelvin-Voigt, Maxwell, and Maxwell-Weichert constitutive models. The models are verified by comparison with experimental data for viscoelastic solids and fluids. We consider uniaxial tension of a bar and radial oscillations of a thick-walled spherical shell made of the fractional Kelvin-Voigt incompressible material. Explicit solutions to these problems are derived and compared with experimental data for styrene butadiene rubber and synthetic rubber. It is shown that the fractional Kelvin-Voigt model provides excellent prediction of experimental data. For uniaxial tension of a bar and simple shear of an infinite layer made of the fractional Maxwell compressible material, we develop explicit solutions and compare them with experimental data for polyisobutylene specimens. It is shown that the fractional Maxwell model ensures fair agreement between experimental data and results of numerical simulation. This model allows the number of adjustable parameters to be reduced significantly compared with other models which ensure the same level of accuracy in the prediction of experimental data.

A model for the nonlinear viscoelastic response in polymers at finite strains

Abstract New constitutive equations are derived for viscoelastic media which do not possess the separability property. The model extends the concept of transient polymeric networks by assuming the rates of breakage and reformation of adaptive links to depend on strain energy density. This permits mechanically induced aging/rejuvenation of polymers to be described. To verify the constitutive relations, we use experimental data obtained in tensile relaxation tests for poly(methyl methacrylate) and polycarbonate, in shear relaxation tests for poly(methyl methacrylate) and an epoxy glass, and in tensile tests with constant rate of strains for ethylene—butene and ethylene-octene copolymers, poly(ethylene terephthalate), and polycarbonate at strains up to 300 per cent. The model provides fair agreement between observations and their predictions. As an example of inhomogeneous deformations, we analyze inflation of a thick-walled viscoelastic spherical vessel under the action of internal pressure. The effect of material parameters on stresses and displacements is studied numerically.

Constitutive equations in finite viscoplasticity of semicrystalline polymers

Abstract Three series of uniaxial tensile tests with constant strain rates are performed at room temperature on isotactic polypropylene and two commercial grades of low-density polyethylene with different molecular weights. Constitutive equations are derived for the viscoplastic behavior of semicrystalline polymers at finite strains. A polymer is treated as an equivalent network of strands bridged by permanent junctions. Two types of junctions are introduced: affine whose micro-deformation coincides with the macro-deformation of a polymer, and non-affine that slide with respect to their reference positions. The elastic response of the network is attributed to elongation of strands, whereas its viscoplastic behavior is associated with sliding of junctions. The rate of sliding is proportional to the average stress in strands linked to non-affine junctions. Stress–strain relations in finite viscoplasticity of semicrystalline polymers are developed by using the laws of thermodynamics. The constitutive equations are applied to the analysis of uniaxial tension, uniaxial compression and simple shear of an incompressible medium. These relations involve three adjustable parameters that are found by fitting the experimental data. Fair agreement is demonstrated between the observations and the results of numerical simulation. It is revealed that the viscoplastic response of low-density polyethylene in simple shear is strongly affected by its molecular weight.

Modelling the viscoplastic response of polyethylene in uniaxial loading¿unloading tests

Abstract Two series of uniaxial cyclic tests are performed on low-density polyethylene at room temperature. In the first series of experiments, injection-molded specimens are stretched to several maximal strains ϵmax in the region of sub-yield deformations with a constant cross-head speed, ϵ =10 mm/min, and retracted down to the zero stress with the same strain rate. In the other series, loading–unloading tests are carried out with the maximal strain ϵmax=0.10 and cross-head speeds ranging from 5 to 200 mm/min. A constitutive model is derived for the viscoplastic behavior of a semicrystalline polymer at small strains. A polymer is modelled as an equivalent network of chains bridged by permanent junctions (entanglements, physical cross-links on the surfaces of crystallites and lamellar blocks). The network is treated as an ensemble of meso-regions connected by links (crystalline lamellae). Deformation of a specimen induces sliding of junctions with respect to their reference positions both at active loading and unloading (this process reflects sliding of junctions in amorphous regions and fine slip of crystalline lamellae). At retraction, sliding of junctions is accompanied by mutual displacements of meso-domains (that reflects coarse slip and fragmentation of lamellar blocks). The constitutive equations are determined by 5 adjustable parameters that are found by matching the experimental stress–strain curves.

Physical aging in amorphous polymers far below the glass transition temperature

Abstract Constitutive equations are derived for the viscoelastic response and enthalpy recovery in amorphous polymers quenched far below the glass transition temperature. The model is based on the concept of cooperative relaxation, which treats a polymer as an ensemble of independent relaxing regions. Any flow unit is trapped in a cage, where it randomly hops in the potential well being thermally activated. Rearrangement occurs, when a relaxing region reaches in a hop some liquid-like state. Structural relaxation in a disordered medium after quenching is thought of as an increase in roughness of the energy landscape. Using plausible hypotheses about the rearrangement process, stress–strain relations are derived which account for the effect of physical aging on the viscoelastic behavior, and a formula is developed for the increment of the specific enthalpy. These relationships are validated using experimental data in mechanical and calorimetric tests on polycarbonate and polystyrene. Fair agreement is demonstrated between observations and results of numerical simulation.